Hypergroups and hypergroup algebras

Litvinov, G. L.

doi:10.1007/bf01088201

Cited by 34 publications

(25 citation statements)

References 111 publications

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“…Recall that the comultiplication, the counit, and the antipode of U(g) are uniquely determined by ∆(X) = X ⊗ 1 + 1 ⊗ X, ε(X) = 0, S(X) = −X (X ∈ g). Many other examples of well-behaved Hopf ⊗-algebras can be found in [41], [42], [43], [1], and [57].…”

Section: Localizations Of U(g) and Dualitymentioning

confidence: 99%

“…By definition, the hyperenveloping algebra F(g) is the strong dual algebra of O e . (Note that the original definition of F(g) given by Rashevskii in [63] was different; we follow the approach suggested by Litvinov [41,43] There is a natural Hopf algebra pairing between U(g) and O e defined as follows (for details, see [55, 3.2]). For each X ∈ g, let X denote the corresponding left-invariant vector field on G. For each open set U ⊂ G we use the same symbol X to denote the corresponding derivation of O(U).…”

Section: Algebras Of Analytic Functionals and Hyperenveloping Algebrasmentioning

confidence: 99%

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Stably flat completions of universal enveloping algebras

Pirkovskii

2006

Dissertationes Math.

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We study localizations (in the sense of J. L. Taylor [70]) of the universal enveloping algebra, U (g), of a complex Lie algebra g. Specifically, let θ : U (g) → H be a homomorphism to some well-behaved [1] topological Hopf algebra H. We formulate some conditions on the dual algebra, H ′ , that are sufficient for H to be stably flat [52] over U (g) (i.e., for θ to be a localization). As an application, we prove that the Arens-Michael envelope, U (g), of U (g) is stably flat over U (g) provided g admits a positive grading. We also show that Goodman's weighted completions [19] of U (g) are stably flat over U (g) for each nilpotent Lie algebra g, and that Rashevskii's hyperenveloping algebra [63] is stably flat over U (g) for arbitrary g. Finally, Litvinov's algebra A (G) of analytic functionals [38,39,41] on the corresponding connected, simply connected complex Lie group G is shown to be stably flat over U (g) precisely when g is solvable.

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Section: Localizations Of U(g) and Dualitymentioning

confidence: 99%

Section: Algebras Of Analytic Functionals and Hyperenveloping Algebrasmentioning

confidence: 99%

Stably flat completions of universal enveloping algebras

Pirkovskii

2006

Dissertationes Math.

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“…There is a considerable literature on hypergroups and hypergroup algebras (for example, see the survey article [12]). We present a new theory of this type but with more precise conditions than previous versions.…”

Section: Introductionmentioning

confidence: 99%

Multivalued Groups, n-Hopf Algebras and n-Ring Homomorphisms

Buchstaber

Rees

1998

Lie Groups and Lie Algebras

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Abstract. This paper discusses the concept of an n-valued group. The theory that is developed here can be regarded as having more precise conditions than those in the much more developed theory of hypergroupsj it is also more oriented towards algebraic and topological ideas. A number of methods for constructing examples are given and the classification of the algebraic 2-valued group structures on the Riemann sphere is presented. We introduce the (purely algebraic) notion of an n-Hopf algebra and show that the ring of functions on an n-valued group has an n-Hopf algebra structure. The crucial property of an n-Hopf algebra is that the diagonal is an n-ring homomorphism that has a weaker multiplicative property than that satisfied by a ring homomorphism. This concept arises in a number of other contexts, in particular as a trace. Convolution of two ring homomorphisms from an n-Hopf algebra to any commutative ring yields an n-ring homomorphism. There is a duality between symmetric products of topological spaces and the set of all n-ring homomorphisms from the ring of functions on the space to C. The development of the properties of n-ring homomorphisms involves a number of combinatorial identities related closely to the classical properties of symmetric polynomials. Mathematics

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“…About twenty years have elapsed since the articles of Dunkl, Jewett and Spector, and in that time the field has grown by attracting workers in special functions, probability, and harmonic analysis. Students of hypergroups relied mainly on Jewett's paper as a text (the lucky ones of us also have a copy of Kenneth Ross's index of that paper) and for more recent developments the surveys of Heyer [Hey84b] and Litvinov [Lit87] (which also treats developments in other systems similar to hypergroups). The field is young and still growing rapidly in many directions, and the book is necessarily a snapshot of its current state.…”

Section: Hypergroups Each Of (M (mentioning

confidence: 99%